How many different sets of four dots
can be joined to form a square?

A printable sheet for student use is available here

Topics: Starter | Combinations | Puzzles | Shape

• Greg, New York
•
• On the website, it says that there are only 4 green ones and that the total is 11. they are wrong. There are actually 8 green ones and the total is 15 because there are 4 big green squares on the outside and 4 small green squares in the center.
• Rob, Estonia
•
• Read the question again - the 4 green squares you point to are not drawn by connecting the dots and thus do not count towards the answer. 11 is correct.
• Year 10 Set 2, Welshpool High School Powys
•
• Only George and Daniel got the right answer. Most thought it was 9.
• Mr Seale's Class 2011, Holywell High School
•
• We think we can find 13, with the 4 extra squares in the middle! :).
• Eddy, Wellington, New Zealand
•
• Cool, I liked this one. I got 9.
• Laura Nortman, United Statesq
•
• Please be certain that your answers are clear and correct. On Answer 11, on the issue of how many squares there are in the image. You stated that there were only 4 green squares (I am assuming those would be the four in the middle). However, those 4 squares can be combined to make one more larger square. You can then take the green rectangle on the top of of the image and combine it with the two small green squares below it, that makes another larger green square. You can follow this same process with the bottom green rectangle and the two side green rectangles. Combine each of the green rectangles with the two adjacent small green rectangles and you will see the square. This means that there are 4 small green squares and 5 additional large green squares...for a total of 16 squares. This of course takes critical thinking, which is something we apparently don't care to pass onto our children anymore. You should not claim to be an educational resource and provide such bad example for our youth. THINK people. If you wanted to clarify, you could also have asked how many squares there were that did not overlap parts of other squares, but I think forcing them to stand back and look at the picture to see the larger squares is a better teaching tool...of course you have to provide the right answer...which you did not.
• John, UK
•
• This is bizarre and all the proposed solutions are wrong. The question asks for you to join 4 dots. The maximum number of squares that are possible with 4 dots is 1. If you join multiple sets of 4 dots which is not specified but what seems to be taken as the instruction for some reason, then taking the colours used lower on the page I can clearly see 9 green squares not 5.
• Hendrik, Edinburgh
•
• Very good. I was able to get nine and then the additional red and blue square when I read the answer was eleven, without looking at the diagram.
I don't wish to be too critical of those commenters who disagree with this total, as I am about to do the same, but it seems to me that their solutions all involve connecting more than four dots.
I can see at least an additional nine possible squares, connecting only four dots. Your solution is correct only for the squares with the dots at the corners, a condition which was not specified in the question.
If you take each of the black squares and treat it as an inscribed square of a square twice the area, at 45 degrees with respect to it, a common centre and with each dot in the _middle_ of a side, not at the corner, that gives you an additional five squares, each only joining four dots. Further, if you take a line through any four dots in a row, you can make that a single side of a large square which extends away from the centre of the diagram. This square also passes through exactly four dots, two at a corner, two in one of the lines but not at a corner and contains one line outside the boundaries of the diagram which contains no dots at all. There are four rows which contain four dots in a line so this gives another additional four squares, making a total of twenty in all.
• Hendrik, Edinburgh
•
• Woops! Actually, there's an infinite number! If you take any two adjacent dots on the outside of the diagram and draw a line through them and then extend that line to meet another line produced in the same way but with two of the other outside dots and at right angles to the first line, that gives you the corner of a square. You can then extend both those perpendicular lines any equal distance you like in the opposite direction, giving you an infinite range of differently sized squares, each passing through exactly four dots. The only caveat is that, while the boundaries of the squares can be inside or outside of those of the diagram, they cannot be of a size to pass through any dots, other than the ones used in their construction.
• John, UK
•
• What it boils down to is a badly worded question. If the question were specified correctly there could be no discussion about the correct answer. My son is currently doing A level physics and because of the inability of the question writers to see the correct answers that other people may see the school has changed the exam board. While this has improved the situation there are still lots of ambiguities in the questions.
• 8Y2, BBS
•
• 8Y2 found 9. We thought there might be 11.
• David McColgan, Decroly Americano, Guatemala
•
• The question is worded perfectly if you take the instructions down the side as part of the question. The only way to form squares using those instructions is by connecting 4 dots which become the 4 vertices of the square. Thanks to whoever made it up. The other squares people have referenced cannot be made on this website by joining exactly 4 dots.
Another way to think about it. Each dot on the outside is a vertex of three different squares (8*3=24). Each dot in the middle is a vertext of 5 different squares (4*5=20). (24+20)/4=11. We concluded this by drawing every possible line on the diagram. There are obvioulsy other squares visible, but not ones which abide by the full instructions.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

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There is a printable worksheet to go with this activity.

## Ideas for Extension Activity

How many rectangles can be found by joining four dots on the grid?

How many triangles can be found by joining three dots on the grid?

How many polygons can be found by joining dots on the grid?

What are the areas of the polygons that can be found?

Assume that the horizontal and vertical distances between the dots
are one unit and the dots are symmetrically positioned.

Here is the URL which will take them to a different type of counting activity.

Transum.org/go/?to=Misfits

Other shape counting starters:

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